MATH 3581
College
Geometry
D. P. Dwiggins, PhD
Department of
Mathematical Sciences
Course Outline
• Part I = Euclidean Geometry
• Circles and Triangles
• Concurrence, Collinearity, and Concyclicity
• Constructions
• Part II = Projective Geometry
• Part III = Noneuclidean Geometry
• History of the Parallel Postulate
• Spherical Geometry and Trigonometry
• Hyperbolic and Elliptic Geometry
Analytic Geometry vs
Synthetic Geometry
Analysis: the act of taking something apart to see how it works
Analytic Geometry: use a coordinate system to assign numbers
to every part of a geometric object, determine which algebraic
equations are used to relate these numbers, and use the equations
to deduce properties of the geometric object
Synthesis: the act of making the parts fit together to form the whole
Synthetic Geometry: begin with a stated list of axioms and
postulates concerning geometric space, and use rules of logic and
the formation of syllogistic arguments to deduce properties of the
objects in geometric space
Analytic Geometry vs
Synthetic Geometry
Beginning in the 1960’s, since analytic geometry is used to
illustrate the development of concepts in calculus, colleges
began referring to their calculus courses as “Calculus and
Analytic Geometry.” However, the only real geometry being
done involved equations for lines, circles, and conic sections.
The intent was for high school students to study synthetic
geometry, basically by following Euclid’s text. Since
college-bound students were also supposed to be reading
Homer and classical texts in Latin, it was anticipated that
first-year college students would be well-based in classical
literature, philosophy, and mathematical development.
This of course never came anywhere close to happening.
For both of these reasons, a new course in College Geometry
was created at the U of M in the early 1990’s, and other
universities across the nation have begun doing the same.
High School
Geometry
A typical high school geometry course outline would be
to work through the material found in Euclid I:1-48,
which begins with the construction of an equilateral
triangle and concludes with the Pythagorean Theorem.
Along the way, triangle congruence theorems are used
to deduce properties of circles and quadrilaterals.
Finally, the postulate of measure is introduced and
students learn how to calculate areas, volumes, and
proportions in geometric figures.
Unfortunately, in modern times the emphasis is on
learning the material needed to answer questions on the
ACT, and most of this material is quickly forgotten once
the test is over.
Geometry
on the ACT
• Definitions
• Lines and Angles
• Polygons and Triangles
• Polygons
• Similar Triangles
• Quadrilaterals
• Areas of Irregularly Shaped Regions
• Solids
• Surface Area and Volume
• Prisms, Pyramids, and Spheres
• Circles
• Central and Inscribed Angles
• Secant and Tangent Lines
• Chords and Arcs
Angles on a Transversal
for Parallel Lines
1
3
5
7
2
4
6
8
Corresponding Angles are Congruent
Vertical Angles are Congruent
Alternate Interior Angles are Congruent
Same-Side Interior Angles are Supplementary
et cetera
Quadrilaterals
Rectangle
Equiangular, perhaps
not equilateral
Rhombus
Equilateral, perhaps
not equiangular
Kite
Diagonals are
perpendicular
Trapezoid
Two sides are parallel,
but not the other two
Circles and Angles
A
80º
Central Angles
Vertex at center
m AB  80
B
P
Inscribed Angles
Vertex on circumference
35º
m PQ  70
Q
Circles and Lines
Tangent Line: AB touches
B
the circle at one point (B)
A
tangere, to touch
D
C
Secant Line: AC cuts the
circle at two points (C , D)
secare, to cut
Euclid III:36 states the circle divides the secant externally
in a product giving the same as the square of the tangent.
That is, AD  AC  AB2 .
Circles and Lines
B
A
Actually, here is how Euclid
III:36 is stated, when translated
from Greek into English:
D
C
If a point be taken outside a circle and from it there fall on the circle two
straight lines, and if one of them cut the circle and the other touch it, the
rectangle contained by the whole of the straight line which cuts the circle
and the straight line intercepted on it outside between the point and the
convex circumference will be equal to the square on the tangent.
In Greek mathematics, the only way they made sense of the product ab
of two positive numbers a and b was to interpret it as the area of a
rectangle with sides of length a and b. This is why the product aa = a2
is called the square of a number.
Harmonic Ratios
A
Here is how Euclid III:35 is stated:
E
If in a circle two straight lines cut one another,
the rectangle contained by the segments of the
one is equal to the rectangle contained by the
segments of the other.
D
C
B
Remembering how the Greeks viewed products in terms of the areas of
rectangles, write down what this statement means in terms of the labeled parts
in the diagram. Why is this statement trivial if E is the center of the circle?
Is it possible to use this result to obtain other results?
For example, is there any way to obtain a relation between AB and CD?
What happens if AB  CD? (See Euclid III:3-4.)